Pricing Scheme REF  follows an economically reasonable idea. The agent chooses the new price of each alternative according to the difference between supply and demand. If supply of an alternative in the last period exceeded its demand, then its new price will be lower than before, and vice versa. As prices have to be nonnegative, the price vector is projected on the nonnegative orthant. The convergence analysis of pricing scheme REF  follows from the analysis of the proximal gradient methods see e. g. [1]}. In fact,
Pricing Scheme REF  coincides with the proximal gradient method with constant stepsize for the problem (REF ). For that, we recall that the function \(TER\)  is convex. Clearly, the inverse of the strong smoothness parameter of \(TER\)  is chosen as the largest possible stepsize. The prox-operator for the indicator function of the nonnegative orthant is simply the projection onto \(\mathbb {R}^n_+\) . The derivation of the gradient of the \(TER\) -function in (REF ) concludes the assertion.
Therefore, the rate of convergence follows from the analysis of the proximal gradient method.
